Phase behavior of polydisperse sticky hard spheres: analytical solutions and perturbation theory

We discuss phase coexistence of polydisperse colloidal suspensions in the presence of adhesion forces. The combined effect of polydispersity and Baxter's sticky-hard-sphere (SHS) potential, describing hard spheres interacting via strong and very short-ranged attractive forces, give rise, within the Percus-Yevick (PY) approximation, to a system of coupled quadratic equations which, in general, cannot be solved either analytically or numerically. We review and compare two recent alternative proposals, which we have attempted to by-pass this difficulty. In the first one, truncating the density expansion of the direct correlation functions, we have considered approximations simpler than the PY one. These $C_{n}$ approximations can be systematically improved. We have been able to provide a complete analytical description of polydisperse SHS fluids by using the simplest two orders $C_{0}$ and $C_{1}$, respectively. Such a simplification comes at the price of a lower accuracy in the phase diagram, but has the advantage of providing an analytical description of various new phenomena associated with the onset of polydispersity in phase equilibria (e.g. fractionation). The second approach is based on a perturbative expansion of the polydisperse PY solution around its monodisperse counterpart. This approach provides a sound approximation to the real phase behavior, at the cost of considering only weak polydispersity. Although a final seattlement on the soundness of the latter method would require numerical simulations for the polydisperse Baxter model, we argue that this approach is expected to keep correctly into account the effects of polydispersity, at least qualitatively.Comment: 12 pages, 4 figures, to appear in Molec. Phys. special issue Liblice 200

On the compressibility equation of state for multicomponent adhesive hard sphere fluids

The compressibility equation of state for a multicomponent fluid of particles interacting via an infinitely narrow and deep potential, is considered within the mean spherical approximation (MSA). It is shown that for a class of models leading to a particular form of the Baxter functions $q_{ij}(r)$ containing density-independent stickiness coefficient, the compressibility EOS does not exist, unlike the one-component case. The reason for this is that a direct integration of the compressibility at fixed composition, cannot be carried out due to the lack of a reciprocity relation on the second order partial derivatives of the pressure with respect to two different densities. This is, in turn, related to the inadequacy of the MSA. A way out to this drawback is presented in a particular example, leading to a consistent compressibility pressure, and a possible generalization of this result is discussed.Comment: 13 pages, no figures, accepted for publication Molec. Physics (2002

Stability boundaries, percolation threshold, and two phase coexistence for polydisperse fluids of adhesive colloidal particles

We study the polydisperse Baxter model of sticky hard spheres (SHS) in the modified Mean Spherical Approximation (mMSA). This closure is known to be the zero-order approximation (C0) of the Percus-Yevick (PY) closure in a density expansion. The simplicity of the closure allows a full analytical study of the model. In particular we study stability boundaries, the percolation threshold, and the gas-liquid coexistence curves. Various possible sub-cases of the model are treated in details. Although the detailed behavior depends upon the particularly chosen case, we find that, in general, polydispersity inhibits instabilities, increases the extent of the non percolating phase, and diminishes the size of the gas-liquid coexistence region. We also consider the first-order improvement of the mMSA (C0) closure (C1) and compare the percolation and gas-liquid boundaries for the one-component system with recent Monte Carlo simulations. Our results provide a qualitative understanding of the effect of polydispersity on SHS models and are expected to shed new light on the applicability of SHS models for colloidal mixtures.Comment: 37 pages, 7 figures, 1 tabl

Structure factors for the simplest solvable model of polydisperse colloidal fluids with surface adhesion

Closed analytical expressions for scattering intensity and other global structure factors are derived for a new solvable model of polydisperse sticky hard spheres. The starting point is the exact solution of the ``mean spherical approximation'' for hard core plus Yukawa potentials, in the limit of infinite amplitude and vanishing range of the attractive tail, with their product remaining constant. The choice of factorizable coupling (stickiness) parameters in the Yukawa term yields a simpler ``dyadic structure'' in the Fourier transform of the Baxter factor correlation function $q_{ij}(r)$, with a remarkable simplification in all structure functions with respect to previous works. The effect of size and stickiness polydispersity is analyzed and numerical results are presented for two particular versions of the model: i) when all polydisperse particles have a single, size-independent, stickiness parameter, and ii) when the stickiness parameters are proportional to the diameters. The existence of two different regimes for the average structure factor, respectively above and below a generalized Boyle temperature which depends on size polydispersity, is recognized and discussed. Because of its analycity and simplicity, the model may be useful in the interpretation of small-angle scattering experimental data for polydisperse colloidal fluids of neutral particles with surface adhesion.Comment: 32 pages, 7 figures, RevTex style, to appear in J. Chem. Phys. 1 December 200

Pathologies in the sticky limit of hard-sphere-Yukawa models for colloidal fluids. A possible correction

A known `sticky-hard-sphere' model, defined starting from a hard-sphere-Yukawa potential and taking the limit of infinite amplitude and vanishing range with their product remaining constant, is shown to be ill-defined. This is because its Hamiltonian (which we call SHS2) leads to an {\it exact}second virial coefficient which {\it diverges}, unlike that of Baxter's original model (SHS1). This deficiency has never been observed so far, since the linearization implicit in the `mean spherical approximation' (MSA), within which the model is analytically solvable, partly {\it masks} such a pathology. To overcome this drawback and retain some useful features of SHS2, we propose both a new model (SHS3) and a new closure (`modified MSA'), whose combination yields an analytic solution formally identical with the SHS2-MSA one. This mapping allows to recover many results derived from SHS2, after a re-interpretation within a correct framework. Possible developments are finally indicated.Comment: 21 pages, 1 figure, accepted in Molecular Physics (2003

Polydisperse fluid mixtures of adhesive colloidal particles

We investigate polydispersity effects on the average structure factor of colloidal suspensions of neutral particles with surface adhesion. A sticky hard sphere model alternative to Baxter's one is considered. The choice of factorizable stickiness parameters in the potential allows a simple analytic solution, within the ``mean spherical approximation'', for any number of components and arbitrary stickiness distribution. Two particular cases are discussed: i) all particles have different sizes but equal stickiness (Model I), and ii) each particle has a stickiness proportional to its size (Model II). The interplay between attraction and polydispersity yields a markedly different behaviour for the two Models in regimes of strong coupling (i.e. strong adhesive forces and low temperature) and large polydispersity. These results are then exploited to reanalyze experimental scattering data on sterically stabilized silica particles.Comment: 9 pages, 2 figures (included), Physica A (2001) to appea

Small Angle Scattering data analysis for dense polydisperse systems: the FLAC program

FLAC is a program to calculate the small-angle neutron scattering intensity of highly packed polydisperse systems of neutral or charged hard spheres within the Percus-Yevick and the Mean Spherical Approximation closures, respectively. The polydisperse system is defined by a size distribution function and the macro-particles have hard sphere radii which may differ from the size of their scattering cores. With FLAC, one can either simulate scattering intensities or fit experimental small angle neutron scattering data. In output scattering intensities, structure factors and pair correlation functions are provided. Smearing effects due to instrumental resolution, vertical slit, primary beam width and multiple scattering effects are also included on the basis of the existing theories. Possible form factors are those of filled or two-shell spheres.Comment: 18 pages, 1 figure, uses elsart.st

Local orientational ordering in fluids of spherical molecules with dipolar-like anisotropic adhesion

We discuss some interesting physical features stemming from our previous analytical study of a simple model of a fluid with dipolar-like interactions of very short range in addition to the usual isotropic Baxter potential for adhesive spheres. While the isotropic part is found to rule the global structural and thermodynamical equilibrium properties of the fluid, the weaker anisotropic part gives rise to an interesting short-range local ordering of nearly spherical condensation clusters, containing short portions of chains having nose-to-tail parallel alignment which runs antiparallel to adjacent similar chains.Comment: 13 pages and 6 figure

Interaction of proteins in solution from small angle scattering: a perturbative approach

In this work, an improved methodology for studying interactions of proteins in solution by small-angle scattering, is presented. Unlike the most common approach, where the protein-protein correlation functions $g_{ij}(r)$ are approximated by their zero-density limit (i.e. the Boltzmann factor), we propose a more accurate representation of $g_{ij}(r)$ which takes into account terms up to the first order in the density expansion of the mean-force potential. This improvement is expected to be particulary effective in the case of strong protein-protein interactions at intermediate concentrations. The method is applied to analyse small angle X-ray scattering data obtained as a function of the ionic strength (from 7 to 507 mM) from acidic solutions of $\beta$-Lactoglobuline at the fixed concentration of 10 $\rm g L^{-1}$. The results are compared with those obtained using the zero-density approximation and show a significant improvement particularly in the more demanding case of low ionic strength.Comment: 12 pages, 3 figures, to appear in Biophysical Journal (April 2002) Due to an unfortunate name mismatch, the original submission contained an incorrect sourc

Patchy sticky hard spheres: analytical study and Monte Carlo simulations

We consider a fluid of hard spheres bearing one or two uniform circular adhesive patches, distributed so as not to overlap. Two spheres interact via a ``sticky'' Baxter potential if the line joining the centers of the two spheres intersects a patch on each sphere, and via a hard sphere potential otherwise. We analyze the location of the fluid-fluid transition and of the percolation line as a function of the size of the patch (the fractional coverage of the sphere's surface) and of the number of patches within a virial expansion up to third order and within the first two terms (C0 and C1) of a class of closures Cn hinging on a density expansion of the direct correlation function. We find that the locations of the two lines depend sensitively on both the total adhesive coverage and its distribution. The treatment is almost fully analytical within the chosen approximate theory. We test our findings by means of specialized Monte Carlo (MC) simulations and find the main qualitative features of the critical behaviour to be well captured in spite of the low density perturbative nature of the closure. The introduction of anisotropic attractions into a model suspension of spherical particles is a first step towards a more realistic description of globular proteins in solution.Comment: 47 pages, 18 figures, to appear on J. Chem. Phy